A refinement of the Parshin symbol for surfaces

Details A refinement of the Parshin symbol for surfaces A refinement of the Parshin symbol for surfaces

2010-02-13

arxiv.org/abs/1002.2698v2

Mathematics

Algebraic Geometry

50 pages; substantially improved;

Scientific paper

On an algebraic curve there are Tate symbols, which satisfy Weil reciprocity law. The analogues in higher dimensions are the Parshin symbols, which satisfy Kato-Parshin reciprocity laws. We give a refinement of the Parshin symbol for surfaces, using iterated integrals in the sense of Chen. The product of the refined symbol over the cyclic permutations of the functions recovers the Parshin symbol. Also, we construct a logarithmic version of the Parshin symbol. We prove reciprocity laws for both the refined symbol and a logarithm of the Parshin symbol.

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